On the Whitney distortion extension problem for $C^m(\mathbb R^n)$ and $C^{\infty}(\mathbb R^n)$ and its applications to interpolation and alignment of data in $\mathbb R^n$

Abstract

In this announcement we consider the following problem. Let $n,m\geq 1$, $U\subset\mathbb R^n$ open. In this paper we provide a sharp solution to the following Whitney distortion extension problems: (a) Let $φ:U\to \mathbb R^n$ be a $C^m$ map. If $E\subset U$ is compact (with some geometry) and the restriction of $φ$ to $E$ is an almost isometry with small distortion, how to decide when there exists a $C^m(\mathbb R^n)$ one-to-one and onto almost isometry $Φ:\mathbb R^n\to \mathbb R^n$ with small distortion which agrees with $φ$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to \mathbb R^n$ away from $E$. (b) Let $φ:U\to \mathbb R^n$ be $C^{\infty}$ map. If $E\subset U$ is compact (with some geometry) and the restriction of $φ$ to $E$ is an almost isometry with small distortion, how to decide when there exists a $C^{\infty}(\mathbb R^n)$ one-to-one and onto almost isometry $Φ:\mathbb R^n\to \mathbb R^n$ with small distortion which agrees with $φ$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to \mathbb R^n$ away from $E$. Our results complement those of [14,15,20] where there, $E$ is a finite set. In this case, the problem above is also a problem of interpolation and alignment of data in $\mathbb R^n$. The material in this paper appears in the memoir [14].

Theorems & Definitions (35)

• Theorem 2.1
• Definition 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• Example 1
• Example 2
• Theorem 4.1
• Definition 5.1